Theorem REF  can be used when considering the parameter regions of true skewness for certain distribution families. As an example, consider a sequence \(X_n\sim \operatorname{Weibull}(k_n,1)\)  of independent Weibull random variables, where \(k_n\uparrow \frac{1}{1-\log 2}\) ; these are truly positively skewed by Theorem REF . It is clear that the distribution functions of the \(X_n\) 's converge uniformly to the distribution function of \(X\sim \operatorname{Weibull}(\frac{1}{1-\log 2},1)\) , and one can show directly that the \(p\) -th moments of the \(X_n\) 's are uniformly bounded for every \(p\ge 1\)  (see, e.g., [1]}). Theorem REF  then implies that \(X\)  is truly non-negatively skewed.
